zbMATH — the first resource for mathematics

Long-time behavior of numerical solutions to nonlinear fractional ODEs. (English) Zbl 1441.65059
Summary: In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A08 Fractional ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
65D25 Numerical differentiation
RODAS; sysdfod; DFOC
Full Text: DOI
[1] A.A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46 (2010) 660-666. · Zbl 1208.35161
[2] J.A.D. Applelby, I. Györi and D.W. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations. J. Diff. Equa. Appl. 12 (2006) 1257-1275. · Zbl 1119.39003
[3] J.C. Butcher, A stability property of implicit Runge-Kutta methods. BIT Numer. Math. 15 (1975) 358-361. · Zbl 0333.65031
[4] J.C. Butcher, Thirty years of G-stability. BIT Numer. Math. 46 (2006) 479-489. · Zbl 1105.65085
[5] J. Cao, C. Li and Y.Q. Chen, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II). Fract. Calc. Appl. Anal. 18 (2015) 735-761. · Zbl 1325.65121
[6] W. Cao, Z. Zhang and G.E. Karniadakis, Time-splitting schemes for fractional differential equations I: smooth solutions. SIAM J. Sci. Comput. 37 (2015) A1752-A1776. · Zbl 1320.65106
[7] W. Cao, F. Zeng, Z. Zhang and G.E. Karniadakis, Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38 (2016) A3070-A3093.
[8] J. Čermák, I. Györi and L. Nechvátal, On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 18 (2015) 651-672. · Zbl 1320.39004
[9] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. 277-285 (2007). · Zbl 1163.45306
[10] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75 (2006) 673-696. · Zbl 1090.65147
[11] G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems. In: Vol. 506 of Numerical Analysis, Lecture Notes in Mathematics. Springer Berlin Heidelberg (1975) 60-74.
[12] G. Dahlquist, G-stability is equivalent to A-stability. BIT Numer. Math. 18 (1978) 384-401. · Zbl 0413.65057
[13] K. Diethelm and N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229-248. · Zbl 1014.34003
[14] K. Diethelm, N.J. Ford and A.D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29 (2002) 3-22. · Zbl 1009.65049
[15] P.P.B. Eggermont, Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line. Math. Comput. 53 (1989) 157-189. · Zbl 0674.65104
[16] L. Galeone and R. Garrappa, On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3 (2006) 565-580. · Zbl 1167.65399
[17] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87 (2010) 2281-2290. · Zbl 1206.65197
[18] R. Garrappa, Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simul. 110 (2015) 96-112.
[19] G.H. Gao, Z.Z. Sun and H.W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014) 33-50. · Zbl 1349.65088
[20] E. Hairer and G. Wanner, Solving ordinary differential equations II, 2nd edition. In: Vol. 14 of Stiff and Differential-Algebraic Equations. Springer Series in Computational Mathematics. Springer, Berlin (1996). · Zbl 0859.65067
[21] J.K. Hale, Asymptotic Behavior of Dissipative Systems. American Mathematical Society, New York (2010).
[22] A.T. Hill, Global dissipativity for A-stable methods. SIAM J. Numer. Anal. 34 (1997) 119-142. · Zbl 0870.65073
[23] A.R. Humphries and A.M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal. 31 (1994) 1452-1485. · Zbl 0807.34057
[24] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36 (2016) 197-221. · Zbl 1336.65150
[25] B. Jin, R. Lazarov, V. Thomée and Z. Zhou, On nonnegativity preservation in finite element methods for subdiffusion equations. Math. Comput. 86 (2017) 2239-2260. · Zbl 1364.65197
[26] B. Jin, B. Li and Z. Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39 (2017) A3129-A3152.
[27] B. Jin, B. Li and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56 (2018) 1-23. · Zbl 1422.65228
[28] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam (2006). · Zbl 1092.45003
[29] N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88 (2019) 2135-2155. · Zbl 1417.65152
[30] H. Li, J. Cao and C. Li, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J. Comput. Appl. Math. 299 (2016) 159-175. · Zbl 1382.65251
[31] C.P. Li and F.R. Zhang, A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193 (2011) 27-47.
[32] Y. Li, Y.Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45 (2009) 1965-1969. · Zbl 1185.93062
[33] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533-1552. · Zbl 1126.65121
[34] C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38 (2016) A2699-A2724.
[35] C. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal. 3 (1983) 439-465. · Zbl 0543.65095
[36] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45 (1985) 463-469. · Zbl 0584.65090
[37] D. Matignon. Stability results for fractional differential equations with applications to control processing. In: Vol. 2 of Computational Engineering in Systems Applications. IMACS, IEEE-SMC, Lille, France (1996) 963-968.
[38] O. Nevanlinna, On the numerical solutions of some Volterra equations on infinite intervals. Math.-Rev. Anal. Numr. Thor. Approx. 5 (1976) 31-57. · Zbl 0356.65116
[39] I. Petras, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press Beijing and Springer-Verlag, Berlin (2011). · Zbl 1228.34002
[40] I. Podlubny, Fractional Differential Equations, Academic Press, London (1998). · Zbl 0922.45001
[41] Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006) 193-209. · Zbl 1094.65083
[42] R. Temam, Infinite dimensional dynamical systems in mechanics and physics. In: Vol. 68 of Applied Mathematical Sciences. Springer-Verlag, Berlin (1998).
[43] D. Wang and A. Xiao, Dissipativity and contractivity for fractional-order systems. Nonlinear Dyn. 80 (2015) 287-294. · Zbl 1345.37086
[44] D. Wang and J. Zou, Dissipativity and contractivity analysis for fractional functional differential equations and their numerical approximations. SIAM J. Numer. Anal. 57 (2019) 1445-1470. · Zbl 1423.34093
[45] Y. Xing and Y. Yan, A higher order numerical method for time fractional partial differential equations with nonsmooth data. J. Comput. Phys. 357 (2018) 305-323. · Zbl 1381.35232
[46] D. Xu, Uniform \(l^1\) behavior for time discretization of a Volterra equation with completely monotonic kernel II: Convergence. SIAM J. Numer. Anal. 46 (2008) 231-259. · Zbl 1170.65104
[47] D. Xu, Decay properties for the numerical solutions of a partial differential equation with memory. J. Sci. Comput. 62 (2015) 146-178. · Zbl 1309.45009
[48] Y. Yan, M. Khan and N.J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56 (2018) 210-227. · Zbl 1381.65070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.